On the representation theory of the symmetry group of the Cantor set

In previous work with Harman, we introduced a new class of representations for an oligomorphic group $G$, depending on an auxiliary piece of data called a measure. In this paper, we look at this theory when $G$ is the symmetry group of the Cantor set. We show that $G$ admits exactly two measures $\mu$ and $\nu$. The representation theory of $(G, \mu)$ is the linearization of the category of $\mathbf{F}_2$-vector spaces, studied in recent work of the author and closely connected to work of Kuhn and Kov\'acs. The representation theory of $(G, \nu)$ is the linearization of the category of vector spaces over the Boolean semi-ring (or, equivalently, the correspondence category), studied by Bouc--Th\'evenaz. The latter case yields an important counterexample in the general theory.